arXiv subjects
Publications and source records attributed to Michael Finkelberg.
The category $\cal{C}$ (studied by Andersen-Jantzen-Soergel) of representations of a Lusztig's small quantum group at a root of unity, together with its modular structure, is defined geometrically, using configuration spaces.
The Semiinfinite Flag Space appeared in the works of B.Feigin and E.Frenkel, and under different disguises was found by V.Drinfeld and G.Lusztig in the early 80-s. Another recent discovery (Beilinson-Drinfeld Grassmannian) turned out to conceal a new incarnation of Semiinfinite Flags. We write down these and other results scattered in folklore. We define the local semiinfinite flag space attached to a semisimple group $G$ as the quotient $G((z))/HN((z))$ (an ind-scheme), where $H$ and $N$ are a Cartan subgroup and the unipotent radical of a Borel subgroup of $G$. The global semiinfinite flag space attached to a smooth complete curve $C$ is a union of Quasimaps from $C$ to the flag variety of $G$. In the present work we use $C=P^1$ to construct the category $PS$ of certain collections of perverse sheaves on Quasimaps spaces, with factorization isomorphisms. We construct an exact convolution functor from the category of perverse sheaves on affine Grassmannian, constant along Iwahori orbits, to the category $PS$. Conjecturally, this functor should correspond to the restriction functor from modules over quantum group with divided powers to modules over the small quantum group.
Let $C$ be a smooth projective curve of genus 0. Let $B$ be the variety of complete flags in an $n$-dimensional vector space $V$. Given an $(n-1)$-tuple $α\in N[I]$ of positive integers one can consider the space $Q_α$ of algebraic maps of degree $α$ from $C$ to $B$. This space admits some remarkable compactifications $Q^D_α$ (Quasimaps), $Q^L_α$ (Quasiflags) constructed by Drinfeld and Laumon respectively. In [Kuznetsov] it was proved that the natural map $π: Q^L_α\to Q^D_α$ is a small resolution of singularities. The aim of the present note is to study the singular support of the Goresky-MacPherson sheaf $IC_α$ on the Quasimaps' space $Q^D_α$. Namely, we prove that this singular support $SS(IC_α)$ is irreducible. The proof is based on the factorization property of Quasimaps' space and on the detailed analysis of Laumon's resolution $π: Q^L_α\to Q^D_α$.
Let $C$ be a smooth projective curve of genus 0. Let $\CB$ be the variety of complete flags in an $n$-dimensional vector space $V$. Given an $(n-1)$-tuple $α\in\BN[I]$ of positive integers one can consider the space $\CQ_α$ of algebraic maps of degree $α$ from $C$ to $\CB$. This space admits some remarkable compactifications $\CQ^D_α$ (Quasimaps), $\CQ^L_α$ (Quasiflags), $\CQ^K_α$ (Stable Maps) of $\CQ_α$ constructed by Drinfeld, Laumon and Kontsevich respectively. It has been proved that the natural map $π: \CQ^L_α\to \CQ^D_α$ is a small resolution of singularities. The aim of the present note is to study the cohomology $H^\bullet(\CQ^L_α,\BQ)$ of Laumon's spaces or, equivalently, the Intersection Cohomology $H^\bullet(\CQ^L_α,IC)$ of Drinfeld's Quasimaps' spaces. We calculate the generating function $P_G(t)$ (``Poincaré polynomial'') of the direct sum $\oplus_{α\in\BN[I]}H^\bullet(\CQ^D_α,IC)$ and construct a natural action of the Lie algebra ${\frak{sl}}_n$ on this direct sum by some middle-dimensional correspondences between Quasiflags' spaces. We conjecture that this module is isomorphic to distributions on nilpotent cone supported at nilpotent subalgebra.
A report on the works hep-th/9411050, q-alg/9412017, q-alg/9503013, q-alg/9506011 and a joint work with R.Bezrukavnikov.